In \triangle ABC, the measure of \angle A is 40^\circ. The measure of \angle B is three ti… — ACT

In $\triangle ABC$ , the measure of $\angle A$ is $40^\circ$ . The measure of $\angle B$ is three times the measure of $\angle C$ . What is the measure, in degrees, of $\angle B$ ?

Answer: 105 degrees

Answer

The measure of angle B is 105 degrees.

The sum of the interior angles in any triangle is

180^\circ

. By setting the measure of

\angle C

x

and the measure of

\angle B

3x

, we can write the equation

40 + 3x + x = 180

. Solving for

x

yields

4x = 140

, which simplifies to

x = 35

. The measure of

\angle B

3x

, which is

3 \times 35 = 105^\circ

Step-by-Step Solution

Set up the equation using the fact that the sum of angles in a triangle is 180 degrees.

m\angle A + m\angle B + m\angle C = 180^\circ

The angles of any triangle in plane geometry sum to 180 degrees.

Represent the unknown angles algebraically.

Let

m\angle C = x

, then

m\angle B = 3x

. Substitute

m\angle A = 40^\circ

This allows solving for the unknown angles with a single-variable equation.

Solve the equation for

x

40 + 4x = 180 \implies 4x = 140 \implies x = 35

To find the measure of angle C.

Calculate the measure of angle B.

m\angle B = 3(35) = 105^\circ

Angle B is three times angle C, and we need to find the measure of angle B.

Key Concept

The sum of the interior angles of a triangle is always 180 degrees.

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